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Why is Euclidean $BTZ$ black hole at temperature $T$, equivalent to thermal $AdS_3$ at temperature $1/T$?

I'm not really familiar with those objects, but I have used the modular transformation

$$\tau_{BTZ} = \frac{-1}{\tau_{AdS}}$$

So, for the same period in spatial length, do they have an opposite temperature? But, what about the $"-"$ sign? Is there any intuition here? Or do you have any references for the link between those two theories?

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